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34-O                                     Geometry.

from plan to meet boiler circumference, and carry these along
horizontally to cross the vertical lines at c D ; the serpentine
curve, being then traced through the numbers obtained, will
represent the developed intersecting line. This may be repeated
on the second half of the plate, and allowances made for flanging
and welding. The boiler hole is developed by stepping-off the
three distances, /$, h^ and H, with dividers, and measuring them
from the vertical centre line in plan to give a, b, and c respec-
tively, the remaining four segments being symmetrical. The
length of plate is found by calculation.

Intersections of oblique cylinder with plane, or cone with
cylinder, are rarely required; but cone with plane is sometimes
necessary, as in funnels for American locomotives, or conical
flues such as that shewn at L, Fig. 310. The latter has been
chosen as an example, and the form of plate developed at K,
Fig. 315, J being the finished flue. The drawing j having been
made, the outer lines are produced to meet at f, and the dotted
circles struck, with gf and//as radii. Upon these are measured
the circumferences at d and e respectively, and allowance made
for welding and flanging.

If the set-squares at hand be not long enough, the marker-off
should be able to set out a right angle by the measurements of
three sides of a triangle, it being easily remembered that the
proportions 3, 4, 5, for base, perpendicular, and hypotenuse in
turn, will serve his purpose, as can be proved by the 47th proposi-
tion of Euclid's first book, thus :

32 +  42=  52      or      9 -f 16 =25

The length of arc, chord being known, is sometimes required,
and may be obtained as follows:

Let *:=the half chord.
r=radius of arc.
a = half the angle subtended by the arc.

Then - = sin a.

r

The angle a being found from a table of sines,

Length of Arc = 2 x - x 2w.r= '0349^.